With spread of digital signal process in recent years, A/D (D/A) converting process is performed in various fields of sound process, image process, communication and the like. Since the precision of the converting process exerts an influence on a result of the sound process and the like, higher level is being continuously demanded. For example, one of methods having high precision is PCM (pulse-code modulation). The PCM, however, has a drawback such that the stability of circuit elements in an A/D converter is insufficient. One of methods realizing stability of operation is ΣΔ (sigma-delta) modulation (refer to Non-Patent Documents 1 to 7). There is also a technique called β conversion based on the sigma-delta modulation (refer to Patent Documents 8 and 9).
The β conversion will be described. When γ=1/β (where 1<β<2), xε(0, 1) is expressed by equation (1) using biε{0, 1}. Here, u1=βy, b1=Qν(u1), and ui+1=β(ui−bi), bi+1=Qν(ui+1). Qν(z) denotes a quantizer and, for a threshold ν satisfying νε[1, (β−1)−1], is 0 when z<ν and is 1 when z≧ν.
                    Equation        ⁢                                  ⁢        1                                                            x        =                              ∑                          i              =              1                        ∞                    ⁢                                    b              i                        ⁢                                          γ                i                            .                                                          (        1        )            
FIG. 8 is a diagram for explaining the structure of a β converter. When z0=yε[0, 1) and i>0, zi=0, and u0=b0=0. In the β converter, when ν=1, the greedy” scheme has been proposed. When ν=(β−1)−1, the “lazy” scheme has been proposed.
When α=ν−1 in the above, it can be said that in the β conversion, the (β, α) expansions proposed by Dajani et al. are performed (refer to Non-Patent Document 10). In a map Nβ,α of the (β, α) expansions with βε(1, 2), when xε[0, (α+1)/β), βx is derived. When xε[(α+1)/β, 1/(β−1)), βx−1 is derived.
FIG. 9(a) is a graph showing the map of the (β, α) expansion, FIG. 9(b) is a graph showing a “greedy” map when ν=1, and FIG. 9(c) is a graph showing a “lazy” map when ν=(β−1)−1.
Non-Patent Document 1    Inose, H., and Yasuda, Y., “A unity bit coding method by negative feedback,” Proceedings of the IEEE, vol. 51, no. 11, pp 1524-1535, November 1963
Non-Patent Document 2    J. Candy, “A Use of Limit Cycle Oscillation to Obtain Robust Analog-to-Digital Converters,” Communications, IEEE Transactions on [legacy, pre-1988], vol. 22, no. 3, pp 298-305, March 1974
Non-Patent Document 3    Stephen H. Lewis, and Paul R. gray, “A pipelined 5-Msample/s 9-bit analog-to-digital converter,” Solid-State Circuits, IEEE Journal of, vol. 22, no. 6, pp 954-961, December 1987
Non-Patent Document 4    Robert M. Gray, “Oversampled Sigma-Delta Modulation,” Communications, IEEE transactions on [legacy, pre-1988], vol. 35, no. 5, pp 481-489, May 1987
Non-Patent Document 5    Robert M. Gray, “Spectral Analysis of Quantization Noise in a Single-Loop Sigma-Delta Modulator with dc Input,” IEEE Transactions on communications, vol. 37, no. 6, pp 588-599, June 1989
Non-Patent Document 6    C. Gunturk, “On the robustness of single-loop sigma-delta modulation,” IEEE Transactions on Information Theory, Vol. 47, no. 5, pp 1734-1744, 2001
Non-Patent Document 7    C. Gunturk, “One-Bit Sigma-Delta Quantization with Exponential Accuracy,” Commun. Pure Applied Math., vol. 56, no. 11, pp 1608-1630, 2003
Non-Patent Document 8    I. Daubechies, R. Devore, C. Gunturk, and V. Vaishampayan, “A/D Conversion With Imperfect Quantizers,” IEEE Transactions on Information Theory, vol. 52, no. 3, pp. 874-885, March 2006
Non-Patent Document 9    I. Daubechies, and O. Yilmaz, “Robust and Practical Analog-to-Digital Conversion With Exponential Precision,” IEEE Transactions on information Theory, vol. 52, no. 8, pp. 3533-3545, August 2006
Non-Patent Document 10    K. Dajani, C. Kraaikamp, “From greedy to lazy expansions and their driving dynamics,” Expo. Math, 2002